Let's show that the derivative of
equals
on any point on the curve.
Thanks to Bryony Miles and his
Titanic scrolling example
that was the technical basis, and Steven Strogatz for his
wonderful book
Infinite Powers
that inspired me to do this.
Reach out to me at
xwhy.graphics _at_ gmail
SCROLL DOWN TO START
Setting the stage
This is our curve. For simplicity, I'm just showing a part of the
curve where x is greater than 0. It looks the same (well, mirrored)
on the other side.
Focus
Now, let's focus on a point of the curve. Grab the point and drag
and drop it along the line, or use left/right arrow keys.
Adding the tanget
Let's now draw the tangent on the selected point. The slope of the
tangent equals the derivative on that point of the curve. The
equation is:
Go on, move the circle
You can still move the circle along our curve. When you're happy
with the location, scroll on.
Zooming in
We're zooming in on this small slice of the curve, 0.5 around our
focus point. See the curve resembling the line in the middle section
(circled).
Enhance!
Zoomed in to 0.1 around our point, the curve and the line are nearly
aligned. Let's see the change in y when we move x just a little
(+-0.01):
Conclusion
By zooming in so much that the curve becomes (virtually) straight
around our point of focus
we see that the rise over run equals
That is, the slope of the tangent in our point of focus is precisely
the value of our derivative (2*x)